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Theorem opi2 4431
Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opi1.1  |-  A  e. 
_V
opi1.2  |-  B  e. 
_V
Assertion
Ref Expression
opi2  |-  { A ,  B }  e.  <. A ,  B >.

Proof of Theorem opi2
StepHypRef Expression
1 prex 4406 . . 3  |-  { A ,  B }  e.  _V
21prid2 3913 . 2  |-  { A ,  B }  e.  { { A } ,  { A ,  B } }
3 opi1.1 . . 3  |-  A  e. 
_V
4 opi1.2 . . 3  |-  B  e. 
_V
53, 4dfop 3983 . 2  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
62, 5eleqtrri 2509 1  |-  { A ,  B }  e.  <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2956   {csn 3814   {cpr 3815   <.cop 3817
This theorem is referenced by:  uniopel  4460  opeluu  4715  elvvuni  4938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823
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